2cos (problem 3.3.5)

Percentage Accurate: 37.6% → 99.2%

Time: 21.7s

Alternatives: 15

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \cos \left(x + \varepsilon\right) - \cos x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

Python

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 37.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(x + \varepsilon\right) - \cos x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

Python

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0033 \lor \neg \left(\varepsilon \leq 0.0037\right):\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos \varepsilon \cdot \cos x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\varepsilon \cdot \sin x\right), \sin x + \cos x \cdot \left(-0.020833333333333332 \cdot {\varepsilon}^{3} + \varepsilon \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0033) (not (<= eps 0.0037)))
   (- (fma (sin eps) (- (sin x)) (* (cos eps) (cos x))) (cos x))
   (*
    -2.0
    (*
     (sin (* 0.5 (+ eps (- x x))))
     (fma
      -0.125
      (* eps (* eps (sin x)))
      (+
       (sin x)
       (*
        (cos x)
        (+ (* -0.020833333333333332 (pow eps 3.0)) (* eps 0.5)))))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0033) || !(eps <= 0.0037)) {
		tmp = fma(sin(eps), -sin(x), (cos(eps) * cos(x))) - cos(x);
	} else {
		tmp = -2.0 * (sin((0.5 * (eps + (x - x)))) * fma(-0.125, (eps * (eps * sin(x))), (sin(x) + (cos(x) * ((-0.020833333333333332 * pow(eps, 3.0)) + (eps * 0.5))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0033) || !(eps <= 0.0037))
		tmp = Float64(fma(sin(eps), Float64(-sin(x)), Float64(cos(eps) * cos(x))) - cos(x));
	else
		tmp = Float64(-2.0 * Float64(sin(Float64(0.5 * Float64(eps + Float64(x - x)))) * fma(-0.125, Float64(eps * Float64(eps * sin(x))), Float64(sin(x) + Float64(cos(x) * Float64(Float64(-0.020833333333333332 * (eps ^ 3.0)) + Float64(eps * 0.5)))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0033], N[Not[LessEqual[eps, 0.0037]], $MachinePrecision]], N[(N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sin[N[(0.5 * N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(eps * N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0033 or 0.0037000000000000002 < eps

   1. Initial program 48.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 99.1%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

      2. sub-neg 99.1%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
   4. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{\left(\left(-\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \cos \varepsilon\right)} - \cos x \]

      2. distribute-lft-neg-in 99.1%

         \[\leadsto \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \cos x \cdot \cos \varepsilon\right) - \cos x \]

      3. *-commutative 99.1%

         \[\leadsto \left(\color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} + \cos x \cdot \cos \varepsilon\right) - \cos x \]

      4. fma-def 99.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon\right)} - \cos x \]

      5. *-commutative 99.1%

         \[\leadsto \mathsf{fma}\left(\sin \varepsilon, -\sin x, \color{blue}{\cos \varepsilon \cdot \cos x}\right) - \cos x \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos \varepsilon \cdot \cos x\right)} - \cos x \]

   if -0.0033 < eps < 0.0037000000000000002

   1. Initial program 27.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. diff-cos 45.7%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      2. div-inv 45.7%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      3. metadata-eval 45.7%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. div-inv 45.7%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

      5. +-commutative 45.7%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval 45.7%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. Applied egg-rr 45.7%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 45.7%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      2. +-commutative 45.7%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      3. associate--l+ 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      4. *-commutative 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

      5. associate-+r+ 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]

      6. +-commutative 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

   5. Simplified 98.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)} \]

   6. Taylor expanded in eps around 0 99.8%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. fma-def 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.125, {\varepsilon}^{2} \cdot \sin x, -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right)\right)}\right) \]

      2. unpow2 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \sin x, -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right)\right)\right) \]

      3. associate-*l* 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \sin x\right)}, -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right)\right)\right) \]

      4. *-commutative 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \color{blue}{\left(\sin x \cdot \varepsilon\right)}, -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + \sin x\right)\right)\right) \]

      5. associate-+r+ 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\sin x \cdot \varepsilon\right), \color{blue}{\left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x}\right)\right) \]

      6. +-commutative 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\sin x \cdot \varepsilon\right), \color{blue}{\sin x + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)}\right)\right) \]

      7. associate-*r* 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\sin x \cdot \varepsilon\right), \sin x + \left(\color{blue}{\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x} + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right) \]

      8. associate-*r* 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\sin x \cdot \varepsilon\right), \sin x + \left(\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x + \color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x}\right)\right)\right) \]

      9. +-rgt-identity 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\sin x \cdot \varepsilon\right), \sin x + \left(\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x + \color{blue}{\left(0.5 \cdot \varepsilon + 0\right)} \cdot \cos x\right)\right)\right) \]

      10. distribute-rgt-out 99.8%

          \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\sin x \cdot \varepsilon\right), \sin x + \color{blue}{\cos x \cdot \left(-0.020833333333333332 \cdot {\varepsilon}^{3} + \left(0.5 \cdot \varepsilon + 0\right)\right)}\right)\right) \]

      11. +-rgt-identity 99.8%

          \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\sin x \cdot \varepsilon\right), \sin x + \cos x \cdot \left(-0.020833333333333332 \cdot {\varepsilon}^{3} + \color{blue}{0.5 \cdot \varepsilon}\right)\right)\right) \]

   8. Simplified 99.8%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\sin x \cdot \varepsilon\right), \sin x + \cos x \cdot \left(-0.020833333333333332 \cdot {\varepsilon}^{3} + 0.5 \cdot \varepsilon\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0033 \lor \neg \left(\varepsilon \leq 0.0037\right):\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos \varepsilon \cdot \cos x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \mathsf{fma}\left(-0.125, \varepsilon \cdot \left(\varepsilon \cdot \sin x\right), \sin x + \cos x \cdot \left(-0.020833333333333332 \cdot {\varepsilon}^{3} + \varepsilon \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0026 \lor \neg \left(\varepsilon \leq 0.0034\right):\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos \varepsilon \cdot \cos x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0026) (not (<= eps 0.0034)))
   (- (fma (sin eps) (- (sin x)) (* (cos eps) (cos x))) (cos x))
   (+
    (-
     (* -2.0 (* (pow eps 3.0) (* (sin x) -0.08333333333333333)))
     (* eps (sin x)))
    (*
     (cos x)
     (+ (* (* eps eps) -0.5) (* (pow eps 4.0) 0.041666666666666664))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0026) || !(eps <= 0.0034)) {
		tmp = fma(sin(eps), -sin(x), (cos(eps) * cos(x))) - cos(x);
	} else {
		tmp = ((-2.0 * (pow(eps, 3.0) * (sin(x) * -0.08333333333333333))) - (eps * sin(x))) + (cos(x) * (((eps * eps) * -0.5) + (pow(eps, 4.0) * 0.041666666666666664)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0026) || !(eps <= 0.0034))
		tmp = Float64(fma(sin(eps), Float64(-sin(x)), Float64(cos(eps) * cos(x))) - cos(x));
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64((eps ^ 3.0) * Float64(sin(x) * -0.08333333333333333))) - Float64(eps * sin(x))) + Float64(cos(x) * Float64(Float64(Float64(eps * eps) * -0.5) + Float64((eps ^ 4.0) * 0.041666666666666664))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0026], N[Not[LessEqual[eps, 0.0034]], $MachinePrecision]], N[(N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0025999999999999999 or 0.00339999999999999981 < eps

   1. Initial program 48.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 99.1%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

      2. sub-neg 99.1%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
   4. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{\left(\left(-\sin x \cdot \sin \varepsilon\right) + \cos x \cdot \cos \varepsilon\right)} - \cos x \]

      2. distribute-lft-neg-in 99.1%

         \[\leadsto \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \cos x \cdot \cos \varepsilon\right) - \cos x \]

      3. *-commutative 99.1%

         \[\leadsto \left(\color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} + \cos x \cdot \cos \varepsilon\right) - \cos x \]

      4. fma-def 99.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \cos \varepsilon\right)} - \cos x \]

      5. *-commutative 99.1%

         \[\leadsto \mathsf{fma}\left(\sin \varepsilon, -\sin x, \color{blue}{\cos \varepsilon \cdot \cos x}\right) - \cos x \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos \varepsilon \cdot \cos x\right)} - \cos x \]

   if -0.0025999999999999999 < eps < 0.00339999999999999981

   1. Initial program 27.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. add-log-exp 27.3%

         \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]

   3. Applied egg-rr 27.3%

      \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 27.3%

         \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

      2. diff-cos 45.7%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      3. +-commutative 45.7%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. associate-+r- 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x - x\right)}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      5. +-inverses 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + \color{blue}{0}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      6. +-commutative 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right)\right) \]

      7. associate-+r+ 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2}\right)\right) \]

   5. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon}}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

   7. Simplified 98.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]

   8. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} \]

      3. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      4. mul-1-neg 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) + \color{blue}{\left(-\varepsilon \cdot \sin x\right)}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      5. unsub-neg 99.8%

         \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) - \varepsilon \cdot \sin x\right)} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(\sin x \cdot \left(-0.0625 + -0.020833333333333332\right)\right)}\right) - \varepsilon \cdot \sin x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      7. metadata-eval 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \color{blue}{-0.08333333333333333}\right)\right) - \varepsilon \cdot \sin x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      8. *-commutative 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \color{blue}{\sin x \cdot \varepsilon}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.5, {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.8%

          \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]

   12. Applied egg-rr 99.8%

       \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0026 \lor \neg \left(\varepsilon \leq 0.0034\right):\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos \varepsilon \cdot \cos x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon \cdot \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0025:\\ \;\;\;\;\left(t_0 - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0034:\\ \;\;\;\;\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos eps) (cos x))))
   (if (<= eps -0.0025)
     (- (- t_0 (* (sin eps) (sin x))) (cos x))
     (if (<= eps 0.0034)
       (+
        (-
         (* -2.0 (* (pow eps 3.0) (* (sin x) -0.08333333333333333)))
         (* eps (sin x)))
        (*
         (cos x)
         (+ (* (* eps eps) -0.5) (* (pow eps 4.0) 0.041666666666666664))))
       (- t_0 (fma (sin eps) (sin x) (cos x)))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) * cos(x);
	double tmp;
	if (eps <= -0.0025) {
		tmp = (t_0 - (sin(eps) * sin(x))) - cos(x);
	} else if (eps <= 0.0034) {
		tmp = ((-2.0 * (pow(eps, 3.0) * (sin(x) * -0.08333333333333333))) - (eps * sin(x))) + (cos(x) * (((eps * eps) * -0.5) + (pow(eps, 4.0) * 0.041666666666666664)));
	} else {
		tmp = t_0 - fma(sin(eps), sin(x), cos(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) * cos(x))
	tmp = 0.0
	if (eps <= -0.0025)
		tmp = Float64(Float64(t_0 - Float64(sin(eps) * sin(x))) - cos(x));
	elseif (eps <= 0.0034)
		tmp = Float64(Float64(Float64(-2.0 * Float64((eps ^ 3.0) * Float64(sin(x) * -0.08333333333333333))) - Float64(eps * sin(x))) + Float64(cos(x) * Float64(Float64(Float64(eps * eps) * -0.5) + Float64((eps ^ 4.0) * 0.041666666666666664))));
	else
		tmp = Float64(t_0 - fma(sin(eps), sin(x), cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0025], N[(N[(t$95$0 - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0034], N[(N[(N[(-2.0 * N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00250000000000000005

   1. Initial program 45.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 99.1%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

   if -0.00250000000000000005 < eps < 0.00339999999999999981

   1. Initial program 27.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. add-log-exp 27.3%

         \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]

   3. Applied egg-rr 27.3%

      \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 27.3%

         \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

      2. diff-cos 45.7%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      3. +-commutative 45.7%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. associate-+r- 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x - x\right)}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      5. +-inverses 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + \color{blue}{0}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      6. +-commutative 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right)\right) \]

      7. associate-+r+ 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2}\right)\right) \]

   5. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon}}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

   7. Simplified 98.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]

   8. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} \]

      3. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      4. mul-1-neg 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) + \color{blue}{\left(-\varepsilon \cdot \sin x\right)}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      5. unsub-neg 99.8%

         \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) - \varepsilon \cdot \sin x\right)} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(\sin x \cdot \left(-0.0625 + -0.020833333333333332\right)\right)}\right) - \varepsilon \cdot \sin x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      7. metadata-eval 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \color{blue}{-0.08333333333333333}\right)\right) - \varepsilon \cdot \sin x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      8. *-commutative 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \color{blue}{\sin x \cdot \varepsilon}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.5, {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.8%

          \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]

   12. Applied egg-rr 99.8%

       \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]

   if 0.00339999999999999981 < eps

   1. Initial program 51.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. sub-neg 51.9%

         \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]

      2. cos-sum 99.0%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]

      3. associate-+l- 98.9%

         \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)} \]

      4. fma-neg 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]
   4. Step-by-step derivation

      1. fma-neg 98.9%

         \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)} \]

      2. *-commutative 98.9%

         \[\leadsto \color{blue}{\cos \varepsilon \cdot \cos x} - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right) \]

      3. *-commutative 98.9%

         \[\leadsto \cos \varepsilon \cdot \cos x - \left(\color{blue}{\sin \varepsilon \cdot \sin x} - \left(-\cos x\right)\right) \]

      4. fma-neg 99.1%

         \[\leadsto \cos \varepsilon \cdot \cos x - \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \sin x, -\left(-\cos x\right)\right)} \]

      5. remove-double-neg 99.1%

         \[\leadsto \cos \varepsilon \cdot \cos x - \mathsf{fma}\left(\sin \varepsilon, \sin x, \color{blue}{\cos x}\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\cos \varepsilon \cdot \cos x - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0025:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0034:\\ \;\;\;\;\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00265 \lor \neg \left(\varepsilon \leq 0.0031\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00265) (not (<= eps 0.0031)))
   (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x))
   (+
    (-
     (* -2.0 (* (pow eps 3.0) (* (sin x) -0.08333333333333333)))
     (* eps (sin x)))
    (*
     (cos x)
     (+ (* (* eps eps) -0.5) (* (pow eps 4.0) 0.041666666666666664))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00265) || !(eps <= 0.0031)) {
		tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x);
	} else {
		tmp = ((-2.0 * (pow(eps, 3.0) * (sin(x) * -0.08333333333333333))) - (eps * sin(x))) + (cos(x) * (((eps * eps) * -0.5) + (pow(eps, 4.0) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00265d0)) .or. (.not. (eps <= 0.0031d0))) then
        tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x)
    else
        tmp = (((-2.0d0) * ((eps ** 3.0d0) * (sin(x) * (-0.08333333333333333d0)))) - (eps * sin(x))) + (cos(x) * (((eps * eps) * (-0.5d0)) + ((eps ** 4.0d0) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00265) || !(eps <= 0.0031)) {
		tmp = ((Math.cos(eps) * Math.cos(x)) - (Math.sin(eps) * Math.sin(x))) - Math.cos(x);
	} else {
		tmp = ((-2.0 * (Math.pow(eps, 3.0) * (Math.sin(x) * -0.08333333333333333))) - (eps * Math.sin(x))) + (Math.cos(x) * (((eps * eps) * -0.5) + (Math.pow(eps, 4.0) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00265) or not (eps <= 0.0031):
		tmp = ((math.cos(eps) * math.cos(x)) - (math.sin(eps) * math.sin(x))) - math.cos(x)
	else:
		tmp = ((-2.0 * (math.pow(eps, 3.0) * (math.sin(x) * -0.08333333333333333))) - (eps * math.sin(x))) + (math.cos(x) * (((eps * eps) * -0.5) + (math.pow(eps, 4.0) * 0.041666666666666664)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00265) || !(eps <= 0.0031))
		tmp = Float64(Float64(Float64(cos(eps) * cos(x)) - Float64(sin(eps) * sin(x))) - cos(x));
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64((eps ^ 3.0) * Float64(sin(x) * -0.08333333333333333))) - Float64(eps * sin(x))) + Float64(cos(x) * Float64(Float64(Float64(eps * eps) * -0.5) + Float64((eps ^ 4.0) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00265) || ~((eps <= 0.0031)))
		tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x);
	else
		tmp = ((-2.0 * ((eps ^ 3.0) * (sin(x) * -0.08333333333333333))) - (eps * sin(x))) + (cos(x) * (((eps * eps) * -0.5) + ((eps ^ 4.0) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00265], N[Not[LessEqual[eps, 0.0031]], $MachinePrecision]], N[(N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00265000000000000001 or 0.00309999999999999989 < eps

   1. Initial program 48.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 99.1%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

   if -0.00265000000000000001 < eps < 0.00309999999999999989

   1. Initial program 27.3%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. add-log-exp 27.3%

         \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]

   3. Applied egg-rr 27.3%

      \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 27.3%

         \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

      2. diff-cos 45.7%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      3. +-commutative 45.7%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. associate-+r- 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x - x\right)}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      5. +-inverses 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + \color{blue}{0}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      6. +-commutative 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right)\right) \]

      7. associate-+r+ 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2}\right)\right) \]

   5. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity 98.5%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon}}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

   7. Simplified 98.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]

   8. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right)\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} \]

      3. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      4. mul-1-neg 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) + \color{blue}{\left(-\varepsilon \cdot \sin x\right)}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      5. unsub-neg 99.8%

         \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(-0.0625 \cdot \sin x + -0.020833333333333332 \cdot \sin x\right)\right) - \varepsilon \cdot \sin x\right)} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(\sin x \cdot \left(-0.0625 + -0.020833333333333332\right)\right)}\right) - \varepsilon \cdot \sin x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      7. metadata-eval 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot \color{blue}{-0.08333333333333333}\right)\right) - \varepsilon \cdot \sin x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

      8. *-commutative 99.8%

         \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \color{blue}{\sin x \cdot \varepsilon}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.5, {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.8%

          \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]

   12. Applied egg-rr 99.8%

       \[\leadsto \left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \sin x \cdot \varepsilon\right) + \cos x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00265 \lor \neg \left(\varepsilon \leq 0.0031\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left({\varepsilon}^{3} \cdot \left(\sin x \cdot -0.08333333333333333\right)\right) - \varepsilon \cdot \sin x\right) + \cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 + {\varepsilon}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.3 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.7e-5) (not (<= eps 4.3e-5)))
   (- (* (cos eps) (cos x)) (+ (cos x) (* (sin eps) (sin x))))
   (- (* -0.5 (* eps (* eps (cos x)))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.7e-5) || !(eps <= 4.3e-5)) {
		tmp = (cos(eps) * cos(x)) - (cos(x) + (sin(eps) * sin(x)));
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.7d-5)) .or. (.not. (eps <= 4.3d-5))) then
        tmp = (cos(eps) * cos(x)) - (cos(x) + (sin(eps) * sin(x)))
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.7e-5) || !(eps <= 4.3e-5)) {
		tmp = (Math.cos(eps) * Math.cos(x)) - (Math.cos(x) + (Math.sin(eps) * Math.sin(x)));
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.7e-5) or not (eps <= 4.3e-5):
		tmp = (math.cos(eps) * math.cos(x)) - (math.cos(x) + (math.sin(eps) * math.sin(x)))
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.7e-5) || !(eps <= 4.3e-5))
		tmp = Float64(Float64(cos(eps) * cos(x)) - Float64(cos(x) + Float64(sin(eps) * sin(x))));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.7e-5) || ~((eps <= 4.3e-5)))
		tmp = (cos(eps) * cos(x)) - (cos(x) + (sin(eps) * sin(x)));
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.7e-5], N[Not[LessEqual[eps, 4.3e-5]], $MachinePrecision]], N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.6999999999999999e-5 or 4.3000000000000002e-5 < eps

   1. Initial program 47.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. sub-neg 47.9%

         \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]

      2. cos-sum 98.6%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]

      3. associate-+l- 98.5%

         \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)} \]

      4. fma-neg 98.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

   3. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

   4. Taylor expanded in x around -inf 98.5%

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)} \]

   if -2.6999999999999999e-5 < eps < 4.3000000000000002e-5

   1. Initial program 27.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left(-\varepsilon \cdot \sin x\right)} \]

      2. unsub-neg 99.8%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) - \varepsilon \cdot \sin x} \]

      3. unpow2 99.8%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) - \varepsilon \cdot \sin x \]

      4. associate-*l* 99.8%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} - \varepsilon \cdot \sin x \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.3 \cdot 10^{-5}\right):\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \left(\cos x + \sin \varepsilon \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 6: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.1 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.4e-5) (not (<= eps 3.1e-5)))
   (- (- (* (cos eps) (cos x)) (* (sin eps) (sin x))) (cos x))
   (- (* -0.5 (* eps (* eps (cos x)))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.4e-5) || !(eps <= 3.1e-5)) {
		tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x);
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.4d-5)) .or. (.not. (eps <= 3.1d-5))) then
        tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x)
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.4e-5) || !(eps <= 3.1e-5)) {
		tmp = ((Math.cos(eps) * Math.cos(x)) - (Math.sin(eps) * Math.sin(x))) - Math.cos(x);
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.4e-5) or not (eps <= 3.1e-5):
		tmp = ((math.cos(eps) * math.cos(x)) - (math.sin(eps) * math.sin(x))) - math.cos(x)
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.4e-5) || !(eps <= 3.1e-5))
		tmp = Float64(Float64(Float64(cos(eps) * cos(x)) - Float64(sin(eps) * sin(x))) - cos(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.4e-5) || ~((eps <= 3.1e-5)))
		tmp = ((cos(eps) * cos(x)) - (sin(eps) * sin(x))) - cos(x);
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.4e-5], N[Not[LessEqual[eps, 3.1e-5]], $MachinePrecision]], N[(N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.4e-5 or 3.10000000000000014e-5 < eps

   1. Initial program 47.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. cos-sum 98.6%

         \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

   3. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x \]

   if -3.4e-5 < eps < 3.10000000000000014e-5

   1. Initial program 27.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left(-\varepsilon \cdot \sin x\right)} \]

      2. unsub-neg 99.8%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) - \varepsilon \cdot \sin x} \]

      3. unpow2 99.8%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) - \varepsilon \cdot \sin x \]

      4. associate-*l* 99.8%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} - \varepsilon \cdot \sin x \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.1 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7: 67.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -5 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -5e-7)
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (* -2.0 (* (sin x) (sin (/ eps 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -5e-7) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else {
		tmp = -2.0 * (sin(x) * sin((eps / 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((eps + x)) - cos(x)) <= (-5d-7)) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = (-2.0d0) * (sin(x) * sin((eps / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -5e-7) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = -2.0 * (Math.sin(x) * Math.sin((eps / 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -5e-7:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = -2.0 * (math.sin(x) * math.sin((eps / 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -5e-7)
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(-2.0 * Float64(sin(x) * sin(Float64(eps / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -5e-7)
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = -2.0 * (sin(x) * sin((eps / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -5e-7], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -4.99999999999999977e-7

   1. Initial program 74.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. diff-cos 74.8%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      2. div-inv 74.8%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      3. metadata-eval 74.8%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. div-inv 74.8%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

      5. +-commutative 74.8%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval 74.8%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      2. +-commutative 74.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      3. associate--l+ 74.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      4. *-commutative 74.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

      5. associate-+r+ 75.0%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]

      6. +-commutative 75.0%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 74.8%

      \[\leadsto -2 \cdot \color{blue}{{\sin \left(0.5 \cdot \varepsilon\right)}^{2}} \]

   if -4.99999999999999977e-7 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 20.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. add-log-exp 20.5%

         \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]

   3. Applied egg-rr 20.5%

      \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 20.5%

         \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

      2. diff-cos 33.4%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      3. +-commutative 33.4%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. associate-+r- 72.0%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x - x\right)}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      5. +-inverses 72.0%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + \color{blue}{0}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      6. +-commutative 72.0%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right)\right) \]

      7. associate-+r+ 72.0%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2}\right)\right) \]

   5. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity 72.0%

         \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon}}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

   7. Simplified 72.0%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]

   8. Taylor expanded in eps around 0 59.4%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\sin x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -5 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array} \]

Alternative 8: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.026:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.001:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.026)
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (if (<= eps 0.001)
     (- (* -0.5 (* eps (* eps (cos x)))) (* eps (sin x)))
     (- (cos eps) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.026) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else if (eps <= 0.001) {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	} else {
		tmp = cos(eps) - cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.026d0)) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else if (eps <= 0.001d0) then
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (eps * sin(x))
    else
        tmp = cos(eps) - cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.026) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else if (eps <= 0.001) {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (eps * Math.sin(x));
	} else {
		tmp = Math.cos(eps) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.026:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	elif eps <= 0.001:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (eps * math.sin(x))
	else:
		tmp = math.cos(eps) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.026)
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	elseif (eps <= 0.001)
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(eps * sin(x)));
	else
		tmp = Float64(cos(eps) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.026)
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	elseif (eps <= 0.001)
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (eps * sin(x));
	else
		tmp = cos(eps) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.026], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.001], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0259999999999999988

   1. Initial program 46.0%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. diff-cos 47.3%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      2. div-inv 47.3%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      3. metadata-eval 47.3%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. div-inv 47.3%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

      5. +-commutative 47.3%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval 47.3%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 47.3%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      2. +-commutative 47.3%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      3. associate--l+ 49.3%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      4. *-commutative 49.3%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

      5. associate-+r+ 49.2%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]

      6. +-commutative 49.2%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

   5. Simplified 49.2%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 49.1%

      \[\leadsto -2 \cdot \color{blue}{{\sin \left(0.5 \cdot \varepsilon\right)}^{2}} \]

   if -0.0259999999999999988 < eps < 1e-3

   1. Initial program 26.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 98.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 98.7%

         \[\leadsto -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left(-\varepsilon \cdot \sin x\right)} \]

      2. unsub-neg 98.7%

         \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) - \varepsilon \cdot \sin x} \]

      3. unpow2 98.7%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) - \varepsilon \cdot \sin x \]

      4. associate-*l* 98.7%

         \[\leadsto -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} - \varepsilon \cdot \sin x \]

   4. Simplified 98.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x} \]

   if 1e-3 < eps

   1. Initial program 52.2%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 54.7%

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.026:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.001:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 9: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* -2.0 (* (sin (/ eps 2.0)) (sin (* 0.5 (+ x (+ eps x)))))))

C

double code(double x, double eps) {
	return -2.0 * (sin((eps / 2.0)) * sin((0.5 * (x + (eps + x)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-2.0d0) * (sin((eps / 2.0d0)) * sin((0.5d0 * (x + (eps + x)))))
end function

Java

public static double code(double x, double eps) {
	return -2.0 * (Math.sin((eps / 2.0)) * Math.sin((0.5 * (x + (eps + x)))));
}

Python

def code(x, eps):
	return -2.0 * (math.sin((eps / 2.0)) * math.sin((0.5 * (x + (eps + x)))))

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(Float64(eps / 2.0)) * sin(Float64(0.5 * Float64(x + Float64(eps + x))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin((eps / 2.0)) * sin((0.5 * (x + (eps + x)))));
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * N[(x + N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Step-by-step derivation

   1. add-log-exp 38.6%

      \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]

3. Applied egg-rr 38.6%

   \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]
4. Step-by-step derivation

   1. add-log-exp 38.6%

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

   2. diff-cos 47.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   3. +-commutative 47.3%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   4. associate-+r- 72.9%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x - x\right)}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   5. +-inverses 72.9%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + \color{blue}{0}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   6. +-commutative 72.9%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right)\right) \]

   7. associate-+r+ 73.0%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2}\right)\right) \]

5. Applied egg-rr 73.0%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity 73.0%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon}}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

7. Simplified 73.0%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]
8. Step-by-step derivation

   1. div-inv 73.0%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \color{blue}{\left(\left(\varepsilon + \left(x + x\right)\right) \cdot \frac{1}{2}\right)}\right) \]

   2. metadata-eval 73.0%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. *-commutative 73.0%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)}\right) \]

   4. add-log-exp 58.2%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\log \left(e^{\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)}\right)}\right) \]

   5. *-commutative 58.2%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \left(e^{\sin \color{blue}{\left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)}}\right)\right) \]

9. Applied egg-rr 58.2%

   \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\log \left(e^{\sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)}\right)}\right) \]
10. Step-by-step derivation

    1. add-log-exp 73.0%

       \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)}\right) \]

    2. associate-+r+ 72.9%

       \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\color{blue}{\left(\left(\varepsilon + x\right) + x\right)} \cdot 0.5\right)\right) \]

    3. +-commutative 72.9%

       \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\color{blue}{\left(x + \left(\varepsilon + x\right)\right)} \cdot 0.5\right)\right) \]

    4. *-commutative 72.9%

       \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)}\right) \]

    5. +-commutative 72.9%

       \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(0.5 \cdot \left(x + \color{blue}{\left(x + \varepsilon\right)}\right)\right)\right) \]

11. Applied egg-rr 72.9%

    \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

12. Final simplification 72.9%

    \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(0.5 \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right) \]

Alternative 10: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* -2.0 (* (sin (/ eps 2.0)) (sin (/ (+ eps (+ x x)) 2.0)))))

C

double code(double x, double eps) {
	return -2.0 * (sin((eps / 2.0)) * sin(((eps + (x + x)) / 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-2.0d0) * (sin((eps / 2.0d0)) * sin(((eps + (x + x)) / 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return -2.0 * (Math.sin((eps / 2.0)) * Math.sin(((eps + (x + x)) / 2.0)));
}

Python

def code(x, eps):
	return -2.0 * (math.sin((eps / 2.0)) * math.sin(((eps + (x + x)) / 2.0)))

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(Float64(eps / 2.0)) * sin(Float64(Float64(eps + Float64(x + x)) / 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin((eps / 2.0)) * sin(((eps + (x + x)) / 2.0)));
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Step-by-step derivation

   1. add-log-exp 38.6%

      \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]

3. Applied egg-rr 38.6%

   \[\leadsto \color{blue}{\log \left(e^{\cos \left(x + \varepsilon\right) - \cos x}\right)} \]
4. Step-by-step derivation

   1. add-log-exp 38.6%

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

   2. diff-cos 47.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   3. +-commutative 47.3%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   4. associate-+r- 72.9%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x - x\right)}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   5. +-inverses 72.9%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + \color{blue}{0}}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   6. +-commutative 72.9%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right)\right) \]

   7. associate-+r+ 73.0%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2}\right)\right) \]

5. Applied egg-rr 73.0%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity 73.0%

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\color{blue}{\varepsilon}}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

7. Simplified 73.0%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)} \]

8. Final simplification 73.0%

   \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

Alternative 11: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.15 \cdot 10^{-20} \lor \neg \left(\varepsilon \leq 2 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.15e-20) (not (<= eps 2e-23)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.15e-20) || !(eps <= 2e-23)) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.15d-20)) .or. (.not. (eps <= 2d-23))) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.15e-20) || !(eps <= 2e-23)) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.15e-20) or not (eps <= 2e-23):
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.15e-20) || !(eps <= 2e-23))
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.15e-20) || ~((eps <= 2e-23)))
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.15e-20], N[Not[LessEqual[eps, 2e-23]], $MachinePrecision]], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.15000000000000006e-20 or 1.99999999999999992e-23 < eps

   1. Initial program 46.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Step-by-step derivation

      1. diff-cos 49.5%

         \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

      2. div-inv 49.5%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(\left(\left(x + \varepsilon\right) - x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      3. metadata-eval 49.5%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot \color{blue}{0.5}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

      4. div-inv 49.5%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

      5. +-commutative 49.5%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \cdot \frac{1}{2}\right)\right) \]

      6. metadata-eval 49.5%

         \[\leadsto -2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \color{blue}{0.5}\right)\right) \]

   3. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(\left(x + \varepsilon\right) - x\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 49.5%

         \[\leadsto -2 \cdot \left(\sin \color{blue}{\left(0.5 \cdot \left(\left(x + \varepsilon\right) - x\right)\right)} \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      2. +-commutative 49.5%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\color{blue}{\left(\varepsilon + x\right)} - x\right)\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      3. associate--l+ 52.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x - x\right)\right)}\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \]

      4. *-commutative 52.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \]

      5. associate-+r+ 52.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \]

      6. +-commutative 52.4%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(x + x\right)\right)}\right)\right) \]

   5. Simplified 52.4%

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)} \]

   6. Taylor expanded in x around 0 51.1%

      \[\leadsto -2 \cdot \color{blue}{{\sin \left(0.5 \cdot \varepsilon\right)}^{2}} \]

   if -2.15000000000000006e-20 < eps < 1.99999999999999992e-23

   1. Initial program 28.4%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 84.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 84.6%

         \[\leadsto \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} \]

      2. mul-1-neg 84.6%

         \[\leadsto \color{blue}{\left(-\varepsilon\right)} \cdot \sin x \]

   4. Simplified 84.6%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.15 \cdot 10^{-20} \lor \neg \left(\varepsilon \leq 2 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 12: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.026:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.026)
   (+ (cos eps) -1.0)
   (if (<= eps 0.00019) (* (sin x) (- eps)) (- (cos eps) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.026) {
		tmp = cos(eps) + -1.0;
	} else if (eps <= 0.00019) {
		tmp = sin(x) * -eps;
	} else {
		tmp = cos(eps) - cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.026d0)) then
        tmp = cos(eps) + (-1.0d0)
    else if (eps <= 0.00019d0) then
        tmp = sin(x) * -eps
    else
        tmp = cos(eps) - cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.026) {
		tmp = Math.cos(eps) + -1.0;
	} else if (eps <= 0.00019) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = Math.cos(eps) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.026:
		tmp = math.cos(eps) + -1.0
	elif eps <= 0.00019:
		tmp = math.sin(x) * -eps
	else:
		tmp = math.cos(eps) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.026)
		tmp = Float64(cos(eps) + -1.0);
	elseif (eps <= 0.00019)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = Float64(cos(eps) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.026)
		tmp = cos(eps) + -1.0;
	elseif (eps <= 0.00019)
		tmp = sin(x) * -eps;
	else
		tmp = cos(eps) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.026], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[eps, 0.00019], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0259999999999999988

   1. Initial program 46.0%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 49.1%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   if -0.0259999999999999988 < eps < 1.9000000000000001e-4

   1. Initial program 26.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 80.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 80.5%

         \[\leadsto \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} \]

      2. mul-1-neg 80.5%

         \[\leadsto \color{blue}{\left(-\varepsilon\right)} \cdot \sin x \]

   4. Simplified 80.5%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]

   if 1.9000000000000001e-4 < eps

   1. Initial program 52.2%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 54.7%

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.026:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 13: 67.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.026 \lor \neg \left(\varepsilon \leq 0.00033\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.026) (not (<= eps 0.00033)))
   (+ (cos eps) -1.0)
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.026) || !(eps <= 0.00033)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.026d0)) .or. (.not. (eps <= 0.00033d0))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.026) || !(eps <= 0.00033)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.026) or not (eps <= 0.00033):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.026) || !(eps <= 0.00033))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.026) || ~((eps <= 0.00033)))
		tmp = cos(eps) + -1.0;
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.026], N[Not[LessEqual[eps, 0.00033]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0259999999999999988 or 3.3e-4 < eps

   1. Initial program 48.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 50.7%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   if -0.0259999999999999988 < eps < 3.3e-4

   1. Initial program 26.8%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in eps around 0 80.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 80.5%

         \[\leadsto \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x} \]

      2. mul-1-neg 80.5%

         \[\leadsto \color{blue}{\left(-\varepsilon\right)} \cdot \sin x \]

   4. Simplified 80.5%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.026 \lor \neg \left(\varepsilon \leq 0.00033\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 14: 45.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.00019\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000155) (not (<= eps 0.00019)))
   (+ (cos eps) -1.0)
   (* (* eps eps) -0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000155) || !(eps <= 0.00019)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000155d0)) .or. (.not. (eps <= 0.00019d0))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = (eps * eps) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000155) || !(eps <= 0.00019)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000155) or not (eps <= 0.00019):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = (eps * eps) * -0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000155) || !(eps <= 0.00019))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(Float64(eps * eps) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000155) || ~((eps <= 0.00019)))
		tmp = cos(eps) + -1.0;
	else
		tmp = (eps * eps) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000155], N[Not[LessEqual[eps, 0.00019]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.55e-4 or 1.9000000000000001e-4 < eps

   1. Initial program 48.6%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 50.3%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   if -1.55e-4 < eps < 1.9000000000000001e-4

   1. Initial program 27.0%

      \[\cos \left(x + \varepsilon\right) - \cos x \]

   2. Taylor expanded in x around 0 27.0%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   3. Taylor expanded in eps around 0 43.1%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 43.1%

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

      2. unpow2 43.1%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 \]

   5. Simplified 43.1%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000155 \lor \neg \left(\varepsilon \leq 0.00019\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 15: 21.0% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) -0.5))

C

double code(double x, double eps) {
	return (eps * eps) * -0.5;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * eps) * (-0.5d0)
end function

Java

public static double code(double x, double eps) {
	return (eps * eps) * -0.5;
}

Python

def code(x, eps):
	return (eps * eps) * -0.5

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) * -0.5)
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * -0.5;
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 38.6%

   \[\cos \left(x + \varepsilon\right) - \cos x \]

2. Taylor expanded in x around 0 39.6%

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

3. Taylor expanded in eps around 0 22.1%

   \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
4. Step-by-step derivation

   1. *-commutative 22.1%

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

   2. unpow2 22.1%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot -0.5 \]

5. Simplified 22.1%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5} \]

6. Final simplification 22.1%

   \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5 \]

Reproduce

herbie shell --seed 2023258 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))